## Tag `0AZT`

Chapter 42: Intersection Theory > Section 42.14: Intersection multiplicities using Tor formula

Lemma 42.14.2. Let $X$ be a nonsingular variety. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules. The $\mathcal{O}_X$-module $\text{Tor}_p^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$ is coherent, has stalk at $x$ equal to $\text{Tor}_p^{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x)$, is supported on $\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G})$, and is nonzero only for $p \in \{0, \ldots, \dim(X)\}$.

Proof.The result on stalks was discussed above and it implies the support condition. The $\text{Tor}$'s are coherent by Lemma 42.14.1. The vanishing of negative $\text{Tor}$'s is immediate from the construction. The vanishing of $\text{Tor}_p$ for $p > \dim(X)$ can be seen as follows: the local rings $\mathcal{O}_{X, x}$ are regular (as $X$ is nonsingular) of dimension $\leq \dim(X)$ (Algebra, Lemma 10.115.1), hence $\mathcal{O}_{X, x}$ has finite global dimension $\leq \dim(X)$ (Algebra, Lemma 10.109.8) which implies that $\text{Tor}$-groups of modules vanish beyond the dimension (More on Algebra, Lemma 15.61.19). $\square$

The code snippet corresponding to this tag is a part of the file `intersection.tex` and is located in lines 722–733 (see updates for more information).

```
\begin{lemma}
\label{lemma-compute-tor-nonsingular}
Let $X$ be a nonsingular variety.
Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_X$-modules.
The $\mathcal{O}_X$-module
$\text{Tor}_p^{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})$
is coherent, has stalk at $x$ equal to
$\text{Tor}_p^{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x)$,
is supported on
$\text{Supp}(\mathcal{F}) \cap \text{Supp}(\mathcal{G})$, and
is nonzero only for $p \in \{0, \ldots, \dim(X)\}$.
\end{lemma}
\begin{proof}
The result on stalks was discussed above and it implies the support
condition. The $\text{Tor}$'s are coherent by
Lemma \ref{lemma-tensor-coherent}. The vanishing of negative
$\text{Tor}$'s is immediate from the construction. The
vanishing of $\text{Tor}_p$ for $p > \dim(X)$ can be seen as follows:
the local rings $\mathcal{O}_{X, x}$ are regular
(as $X$ is nonsingular) of dimension $\leq \dim(X)$
(Algebra, Lemma \ref{algebra-lemma-dimension-prime-polynomial-ring}),
hence $\mathcal{O}_{X, x}$ has finite global dimension $\leq \dim(X)$
(Algebra, Lemma \ref{algebra-lemma-finite-gl-dim-finite-dim-regular})
which implies that $\text{Tor}$-groups of modules vanish beyond the dimension
(More on Algebra, Lemma \ref{more-algebra-lemma-finite-gl-dim-tor-dimension}).
\end{proof}
```

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